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\(x\)
\(+5\)

Write Expressions With
Numbers & Variables

Learn to translate real-world situations into algebraic expressions using numbers, variables, and operation symbols.

6th Grade Math

What Is an Expression?

An expression is a math phrase that combines numbers, variables (letters that stand for unknown numbers), and operations like addition, subtraction, multiplication, and division. Unlike an equation, an expression has no equals sign.

Key Vocabulary

Variable
A letter (like \( x \), \( n \), or \( t \)) that represents an unknown number
Coefficient
The number multiplied by a variable: in \( 5x \), the coefficient is 5
Constant
A number on its own with no variable: in \( 5x + 3 \), the constant is 3

Why Variables Matter

Variables let us write one expression that works for many different numbers. Instead of writing separate calculations for every situation, we use a letter to hold the place of any value.

Without Variables
\(5 + 3 = 8\)
\(5 + 10 = 15\)
\(5 + 27 = 32\)
 ... endless list simplify With a Variable
\(5 + n\)
 works for ANY n

Words That Signal Operations

Operation Key Words Symbol
Addition sum, plus, more than, increased by \( + \)
Subtraction difference, minus, less than, decreased by \( - \)
Multiplication product, times, of, each \( \times \) or \( \cdot \)
Division quotient, divided by, per, split \( \div \) or \( / \)

The Translation Rule

1. Identify the unknown → pick a variable 2. Find key operation words in problem 3. Write the expression using symbols

Multiplication Notation

In algebra, we usually skip the multiplication sign. Writing a number right next to a variable means "multiply."

\( 3 \times x \)
We write this as \( 3x \): the number (coefficient) goes first, then the variable
\( 1 \times n \)
We write this as just \( n \): a coefficient of 1 is invisible

Expression vs Equation

Expression Equation
Equals sign? No Yes
Example \( 2x + 5 \) \( 2x + 5 = 11 \)
What it does Describes a calculation States two things are equal
Can you solve it? Evaluate for a given value Find the unknown value

Step-by-Step Approach

1
Read carefully. Underline the unknown quantity and the operation words.
2
Choose a variable. Let a letter represent the unknown. State what it stands for.
3
Match keywords to symbols. "Sum" → \( + \), "product" → \( \times \), "difference" → \( - \), "quotient" → \( \div \).
4
Write the expression. Combine the numbers, variable, and operation in the correct order.
5
Check with a test value. Plug in a simple number to see if the expression matches the words.

Example 1: Addition

Unknown
\(s\)
 stickers she had
\(+\)
 "12 more"
\(12\)
\(=\)
 Expression
\(s + 12\)

Example 2: Multiplication

8 8 ...
\(b\) boxes
 Expression
\(8b\)

Example 3: Two Operations

1
Unknown: ticket price → variable \( d \)
2
2 tickets: "2 times \( d \)" → \( 2d \)
3
Plus popcorn: "add 5" → \( + 5 \)
✓
Expression: \( 2d + 5 \)

Subtraction & Division Examples

Subtraction
"7 fewer than \( y \)" → \( y - 7 \)

"A number decreased by 4" → \( n - 4 \)
Division
"Split \( m \) equally among 6" → \( \frac{m}{6} \)

"The quotient of 20 and \( p \)" → \( \frac{20}{p} \)

Using Parentheses

Sometimes a word problem tells you to do one operation first. Use parentheses to show which part to calculate before the rest.

Without Parentheses
"3 times \( x \) plus 4" → \( 3x + 4 \)
Multiply first, then add
With Parentheses
"3 times the sum of \( x \) and 4" → \( 3(x + 4) \)
Add first, then multiply

Common Mistakes

✕ Wrong Order
Writing "5 less than \( x \)" as \( 5 - x \) instead of the correct \( x - 5 \). "Less than" reverses the order.
✕ Missing Parentheses
Writing "twice the sum of \( n \) and 3" as \( 2n + 3 \) instead of \( 2(n + 3) \). The word "sum" groups the addition first.
✕ Using \( \times \) Sign
Writing \( 4 \times x \) instead of \( 4x \). In algebra we drop the multiplication sign between a number and variable.

Quick Reference Table

English Phrase Expression
a number plus 9 \( n + 9 \)
12 less than a number \( n - 12 \)
the product of 7 and a number \( 7n \)
a number divided by 4 \( \frac{n}{4} \)
5 more than twice a number \( 2n + 5 \)
half the sum of a number and 6 \( \frac{n + 6}{2} \)

Question 1 of 5

Which expression represents "the sum of a number \( x \) and 15"?

Question 2 of 5

Write an expression for "8 less than \( y \)".

Question 3 of 5

A bag of apples costs \( c \) dollars. Tom buys 4 bags and uses a \$3 coupon. Which expression gives the amount he pays?

Question 4 of 5

Which expression represents "three times the sum of \( n \) and 2"?

Question 5 of 5

\( k \) students share 24 pencils equally. Which expression tells how many pencils each student gets?